# How do you find the volume of the pyramid bounded by the plane 2x+3y+z=6 and the coordinate plane?

the normal vector is

we can re-write the plane as

for

#z= 0, x = 0 implies y = 2# #z= 0, y = 0 implies x = 3#

and

- -

it's this:

the volume we need is

By signing up, you agree to our Terms of Service and Privacy Policy

6

We are going to be performing a triple integral. The cartesian coordinate system is the most applicable. The order of integration is not critical. We are going to go z first, y middle, x last.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the volume of the pyramid bounded by the plane (2x + 3y + z = 6) and the coordinate plane, you first need to determine the height of the pyramid. The height is the perpendicular distance from the apex of the pyramid (the point where the three edges meet) to the base.

The equation of the plane can be rewritten in the form (z = 6 - 2x - 3y). Since the plane intersects the coordinate axes, you can find the intercepts by setting (x = 0), (y = 0), and (z = 0) respectively.

Setting (x = 0), you find (z = 6), which is the intercept on the z-axis. Setting (y = 0), you find (z = 6 - 2x), which gives the intercept on the x-axis. Setting (z = 0), you find (y = 2), which gives the intercept on the y-axis.

From the intercepts, you can determine that the base of the pyramid is a triangle with vertices at (0, 0, 6), (0, 2, 0), and ((3, 0, 0)).

To find the height of the pyramid, you need to find the distance from the apex to the base. This can be calculated using the distance formula or by finding the distance between the apex and a point on the base.

Using the distance formula, the height (h) is given by (h = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}), where ((x_1, y_1, z_1)) is the apex and ((x_2, y_2, z_2)) is a point on the base.

Substituting the coordinates, you can calculate the height.

Once you have the height and the area of the base triangle, you can use the formula for the volume of a pyramid:

[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} ]

Substitute the values to find the volume of the pyramid.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the volume of the region bounded by the graph of #y = x^2+1# for x is [1,2] rotated around the x axis?
- How do you find the volume of a pyramid using integrals?
- The region under the curves #y=e^(1-2x), 0<=x<=2# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
- How do you determine the area of a region above the x-axis and below #f(x)=3+2x-x^2#?
- How do you find the area between #f(x)=3(x^3-x)# and #g(x)=0#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7